Question

Calculate the correlation coefficient from the following data and interpret it.

x	9	7	6	8	9	6	7
y	19	17	16	18	19	16	17

Answer 1

	xi	yi	xi2	yi2	xiyi
	9	19	81	361	171
	7	17	49	289	119
	6	16	36	256	96
	8	18	64	324	144
	9	19	81	361	171
	6	16	36	256	96
	7	17	49	289	119
Total	52	122	396	2136	916

From the table, we have
n = 7, `sum"x"_"i"` = 52, `sum"y"_"i"` = 122, `sum"x"_"i"^2` = 396, `sum"y"_"i"^2` = 2136, `sum"x"_"i""y"_"i"` = 916

∴ `bar"x" = (sum"x"_"i")/"n" = 52/7`

`bar"y" = (sum"y"_"i")/"n" = 122/7`

∴ `bar"x"bar"y"=(52xx122)/49` = `6344/49`

Cov (X, Y) = `1/"n" sum"x"_"i""y"_"i" - bar"x" bar"y"`

= `916/7-6344/49`

= `(6412-6344)/49`

= `68/49`

`sigma_"x"^2 = (sum"x"_"i"^2)/"n" - (bar"x")^2`

= `396/7 - (52/7)^2`

= `(8772-2704)/49`

= `68/49`

`sigma_"y"^2 = (sum"y"_"i"^2)/"n" - (bar"y")^2`

= `2136/7 - (122/7)^2`

= `(14952 - 14884)/49`

= `68/49`

∴ `sigma_"x" sigma_"y" = sqrt(sigma_"x"^2 sigma_"y"^2)`

= `sqrt(68/49 xx 68/49)`

= `68/49`

r = `("Cov (X, Y)")/(sigma_"x" sigma_"y")`

= `((68/49))/(68/49)` = 1
∴ The value of r indicates perfect positive correlation between x and y.